Perovskite oxides form a fascinating class of materials because they possess many active degrees of freedom that result in a large variety of physical effects. One important structural parameter controlling the behavior of perovskites is the tilting of the oxygen octahedral. Among other properties, this tilting is coupled with the electric and magnetic orders, which leads to novel and potentially useful phenomena; recent examples include new mechanisms for improper and triggered ferroelectricity, rich phase diagrams, and novel chiral phases, counter-intuitive behaviors of ferroelectric and multiferroic films, and weak ferromagnetism in otherwise antiferromagnetic materials. Interestingly, most perovskites present the same tilted structures, which are few in number and fairly simple. In contrast, here we use different theoretical methods to show that a complete new family of stable phases, all displaying complex and nano-twinned tilting patterns (as well as other anomalous properties), exists in multiferroic BiFeO3 and related compounds.